Ratios and proportions
Ratio and Proportion Ratio and Proportion
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- Let's start with a warm up ratio problem.
- Then we can tackle some harder word problems. So I have the
- ratio 13/6 is equal to 5/x.
- I don't like having this x in the denominator, so let's
- multiply both sides of this equation by x.
- So if I multiply both sides by x, what's going to happen?
- On the right hand side, this x cancels out with that x.
- And then the left hand side going to become 13 over 6x is
- equal to-- you're just going to have a 5 there.
- And then to solve for x, you just multiply both sides by
- the inverse of 13/6.
- 6/13.
- These, obviously, cancel out.
- That's why I multiplied it by the inverse.
- And you get x is equal to 5 times 6, which is 30/13.
- Now one way that you might see this done-- it's kind of
- skipping a step-- is called cross multiplying.
- You look at a ratio like this, and you immediately say the
- numerator on this side times the denominator on that side
- is equal to the numerator on this side times the
- denominator on that side.
- Let me write that out.
- So you might sometimes see people immediately go to-- let
- me just rewrite the problem actually-- So that original
- problem was 13/6 is equal to 5/x.
- You might sometimes immediately see someone go to
- 13 times x is equal to 5 times 6.
- And it might look like magic.
- How does that work out?
- Why does that make sense?
- And really, all they're doing to get to this point is they
- are simultaneously multiplying both sides of the equation by
- both denominators.
- Let me show you what I mean.
- If I multiply both sides of this equation by 6 and an x,
- what's going to happen?
- If I multiply it by 6x times both sides of this equation--
- And where did I get the 6?
- From here.
- Where did I get the x?
- From there.
- Both denominators.
- What's going to happen?
- On this side of the equation, the 6 is going to cancel out
- with this denominator.
- And on the right hand side of the equation, the x is going
- to cancel with this denominator.
- So you're going to be left with 13 times x is
- equal to 5 times 6.
- So nothing fancy there.
- You're just multiplying by the denominators of both sides of
- the equation.
- And it looks like you're cross multiplying.
- 13x is equal to 5 times 6.
- And then from here, of course, you divide both sides by 13.
- You get x is equal to 30/13.
- Now that we're all warmed up, let's tackle some actual word
- problems.
- So we have the highest mountain in
- Canada is Mount Yukon.
- It is 298/67 the size of then Ben Nevis.
- Let's Y for Yukon is equal to 298/67 the size of-- let's say
- N for Nevis.
- That's what this in green tells us.
- The highest peak in Scotland.
- Mount Elbert in Colorado is the highest peak
- in the Rocky Mountains.
- Mount Elbert-- so we have this other information here-- Mount
- Elbert is 220/67 the height of Ben Nevis.
- So let's say, E for Elbert.
- E is equal to 22/67 times Nevis.
- Times the same Ben Nevis, right there.
- And they're telling us more.
- And, it is 44/48 the size of Mont Blanc.
- So Elbert is equal to 44/48 the size of Mont Blanc.
- Let's write B for Mount Blanc.
- They also tell us Mont Blanc is 4,800 meters high.
- Mont Blanc is 4,800.
- meters high.
- So B is equal to 4,800.
- And they ask us, how high is Mount Yukon?
- So we have to figure out Y.
- So let's see if we can work backwards, and figure out all
- the variables in between.
- So let's start with this information here.
- B is equal to 4,800.
- E is equal to 44/48 times B.
- So E-- so Elbert-- is equal to 44/48 times Mont Blanc, which
- is 4,800 meters.
- Now if you divide that by 48-- 4,800 divided by 48 is 100.
- So Elbert is 44 times 100 meters high.
- So it's equal to 4,400 meters.
- Fair enough.
- Now we can use this information and
- substitute it over here.
- We get Elbert, which is 4,400 meters high, is equal to
- 220/67 times Ben Nevis.
- N for Nevis.
- To solve for Nevis, we multiply both sides by the
- inverse of this coefficient right here.
- So we multiply both sides by 67/220.
- So times 67/220.
- The 67 cancels with that 67.
- That 220 cancels with that 220.
- And then you get-- let's see, if I take 4,400 divided by
- 220-- 440 divided by 220 is 2.
- So this is going to be 20.
- So 4,400 divided by 220 is just 20.
- So you get Nevis is equal to-- I'll swap sides.
- So Ben Nevis is equal to 67 times 20 meters.
- And now that's what?
- 1,340 meters.
- Is that is right?
- Well, lets just leave it like that, because we could--
- actually it looks like that's 67-- I'm going to leave Nevis
- as 67 times 20 meters.
- And substitute it right there.
- So Yukon-- I'll just go down here, because I have more real
- estate there-- Yukon is equal to 298 over 67 times the
- height of Nevis.
- Nevis is 67 times 20.
- So times 67 times 20.
- Well I can divide 67 by 67, and I get Yukon is
- 298 times 20 meters.
- So Yukon is equal to 298 times 20.
- And what is that equal to?
- That is equal to-- Let's see that's 2 times 298
- is going to be 396.
- Oh sorry, it's going to be 596.
- This is almost 300, so it should be close to 600.
- This is 2 less than 300, so this should
- be 4 less than 300.
- And then, I have a 0 here.
- So it's going to be 5,960 meters.
- And we are done.
- Let's do one more of these word problems. All right.
- At a large high school, it is estimated that 2 out of every
- 3 students have a cell phone.
- And 1 in 5 of all students have a cellphone that is one
- year old or less.
- All right.
- So let's think about it.
- Let's say that x is equal to the total number of students.
- This first line, 2 out of 3 students have a cell phone, so
- we could say that 2/3 x have cell phone.
- That's what that green statement tells us.
- And then that purple statement-- 1 in 5 of all
- students have a cellphone that is one year old or less.
- So 1/5 x have less than 1/5 year cell phone.
- So they want to know, out of the students who own a cell
- phone-- so it's out of this-- that's our denominator.
- So let me write that down.
- That is our denominator.
- So out of the students who have a cellphone-- that's
- right there-- they want to know what proportion owns a
- phone that is more than one year old.
- So how many students have a cell phone that is more than
- one year old?
- Well, we could take the total number that have a cellphone,
- which is 2/3 x.
- 2/3 of all the students have a cell phone.
- We can subtract out all of the students that have a new
- cellphone-- a cell phone that is less than one year.
- Remember they're saying more than one year here.
- So we want to subtract out all the students with the new
- cellphone, minus 1/5x, and you will then have the proportion
- of students who have this right here.
- This is right here.
- This is, have greater than 1/5 year cell phone.
- They have a phone, but it's more than 1/5 years old.
- This is all of them that have a cellphone.
- We subtract out the new ones.
- So this is, essentially, all of the people who have an
- older than one-year old cell phone.
- So to solve this, we just subtract the fractions.
- So this is just going to be, let's see, 2/3 is the same
- thing as 10/15.
- That's 2/3 minus 1/5.
- The same thing as 3/15 x.
- Which is equal to 10 minus 3 is 7/15 x.
- Is the total proportion of students-- that's this
- orange-- what proportion owns a phone that is more
- than one year old?
- It's 7/15 x.
- That's an actual number.
- So if you want to know, out of the students who own a cell
- phone-- so out of the students who own a cell phone, right
- there-- 2/3x, what proportion owns a phone that it is more
- than one year old?
- This is the number that own a cellphone that is more than
- one year old.
- And this whole value is the proportion out of the students
- who have a cell phone.
- Lucky for us, the x's cancel out.
- And we are left with this is equal to 7/15 times the
- inverse of the denominator.
- You divide by 2/3.
- That's the same thing as multiplying by 3/2.
- And what does this equal to?
- Divide by 3.
- We are left with 7/10.
- So of the students who own a cell phone, 7 out of 10 of the
- students who own a cell phone, own a cell phone that is more
- than one year old.
- And we are done.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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